The large charge limit of scalar field theories, and the Wilson-Fisher fixed point at 𝜖 = 0

Abstract

We study the sector of large charge operators ϕn (ϕ being the complexified scalar field) in the O(2) Wilson-Fisher fixed point in 4−E dimensions that emerges when the coupling takes the critical value g ∼ 𝜖. We show that, in the limit g → 0, when the theory naively approaches the gaussian fixed point, the sector of operators with n → ∞ at fixed g n2≡ λ remains non-trivial. Surprisingly, one can compute the exact 2-point function and thereby the non-trivial anomalous dimension of the operator ϕn by a full resummation of Feynman diagrams. The same result can be reproduced from a saddle point approximation to the path integral, which partly explains the existence of the limit. Finally, we extend these results to the three-dimensional O(2)-symmetric theory with (ϕ¯ ϕ)3 potential.